Prove that sin36sin72sin108sin144=5/16
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sin36°sin72°sin108°sin144°
=sin36°sin72°sin(180°-72°)sin(180°-36°)
=sin36°sin72°sin{(90°×2)-72°}sin{(90°×2)-36°}
=sin36°sin72°sin72°sin36°
=sin²36°sin²72°
=[√(10-2√5)/4]²×[√(10+2√5)/4]²
=(10-2√5)/16×(10+2√5)/16
=(100-20)/(16×16)
=80/(16×16)
=5/16 (Proved)
=sin36°sin72°sin(180°-72°)sin(180°-36°)
=sin36°sin72°sin{(90°×2)-72°}sin{(90°×2)-36°}
=sin36°sin72°sin72°sin36°
=sin²36°sin²72°
=[√(10-2√5)/4]²×[√(10+2√5)/4]²
=(10-2√5)/16×(10+2√5)/16
=(100-20)/(16×16)
=80/(16×16)
=5/16 (Proved)
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